It is important to emphasize that sampling from any of the radial GF does not completely determine the updated 3D positions [1]}. Rather, to this purpose, one has to use the full solutions describing the angle dependency also. The major drawback is that these full solutions are given by quite unwieldy integrals, which make their evaluation painfully slow. Although, in many situations, the algorithm as a whole is still much more efficient than BD, because large steps can be made, the position update mechanism represents a major bottleneck.
The analytical representation for the full GF describing the diffusion of a point particle around a partially absorbing sphere in 3D is known [2]} to be
\(&&p_{\text{rad}}(r, \theta , t \vert r_{0}) = \tfrac{1}{4\pi \sqrt{r r_{0}}}\sum ^{\infty }_{n = 0}(2n+1)P_{n}(\cos (\theta )) \times \nonumber \\&&\int ^{\infty }_{0}e^{-Dt x^{2}} F_{n+1/2}(r, x) F_{n+1/2}(r_{0}, x)x\,dx.\) 
